Re: Intermediate Value Theorem

Rolles theorem requires the curve to be differentiable (i.e. smooth) not just continuous. MathsIsFuns example is about continuous routes, not smooth ones. You could e.g. walk up the side of a pyramid to the apex and down the another side  in which case no part of your route is horizontal.

Re: Intermediate Value Theorem

The IVT says a lot more than "you will be at the same height". If you walk in a circle (this works for any parametrized continuous path, but circle is easiest to see), let your height at time t be described by h(t). Here we are walking around the circle in 1 unit time of time. Then there will exist at least one point which will be exactly the same height as the opposite side of the circle.

We can define a function:

g(t) = h(t) - h(t+1/2)

If g(0) = 0, then we're done: our starting point is at exactly the same height as the opposite side. Otherwise, g(0) is either positive or negative. But g(1/2) has exactly the opposite sign of g(0):

g(0) = h(0) - h(1/2)g(1/2) = h(1/2) - h(0)

(Remember that h(0) = h(1). They are exactly the same point!)

Now the IVT says that g(t) must be zero at some point in between 0 and 1/2. And that's it.

Of course, probably too difficult to put in the page...

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

Re: Intermediate Value Theorem

Hi all;

MathsisFun wrote:

But the "round trip" example was my own invention and is thus likely to be wrong.

We could argue for awhile about why it is "likely to be wrong" just because it is your own invention. True, that example does have some kinks in it, but does that mean there was a greater than 50% chance that it was an error from the start.

I thought the table problem was much more interesting, just because I have never seen even the handiest people level a table like that.

In mathematics, you don't understand things. You just get used to them.If it ain't broke, fix it until it is.No great discovery was ever made without a bold guess.

Re: Intermediate Value Theorem

bobbym wrote:

True, that example does have some kinks in it, but does that mean there was a greater than 50% chance that it was an error from the start.

Kinks? What Kinks?

Ricky wrote:

The IVT says a lot more than "you will be at the same height". If you walk in a circle (this works for any parametrized continuous path, but circle is easiest to see), let your height at time t be described by h(t). Here we are walking around the circle in 1 unit time of time. Then there will exist at least one point which will be exactly the same height as the opposite side of the circle.

I have attempted to illustrate that ... please have a look at the revised page (at bottom).

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman